1 Solving the Open Vehicle Routing Problem: New Heuristic and Test Problems Feiyue Li Bruce Golden Edward Wasil INFORMS San Francisco November 2005

2 Introduction  Open Vehicle Routing Problem (OVRP) A vehicle does not return to the depot after servicing the last customer on a route Each route in the OVRP is a Hamiltonian path  Two objectives Minimize the total number of vehicles Minimize the total distance traveled We believe the cost of an extra vehicle far exceeds the reduction in distance that can be achieved by an additional route

3 Introduction  Real-world applications of the OVRP FedEx generates incomplete delivery routes for airplanes (Bodin et al. 1983) FedEx Home Delivery service to residential-only customers (Levy 2005) Newspaper home delivery problem (Levy 2005) If a company contracts drivers with vehicles and drivers are not required or paid to return to the depot, then the application fits the OVRP framework

4 Literature Review  Algorithms for the OVRP Since 2000, seven algorithms have been developed to solve OVRP Two use threshold accepting Three use tabu search One uses large neighborhood search One uses the minimum spanning tree

5 Literature Review  Seven algorithms for the OVRP Sariklis and Powell (2000) Cluster First, Route Second (CFRS) Brandao (2004) Tabu Search Algorithm (TSA) Tarantilis, Diakoulaki, and Kiranoudis (2004) Adaptive Memory-based Tabu Search BoneRoute (BR) Tarantilis, Ioannou, Kiranoudis, and Prastacos (2004) Backtracking Adaptive Threshold Accepting (BATA)

6 Literature Review  Seven algorithms for the OVRP Tarantilis, Ioannou, Kiranoudis, and Prastacos (2005) List-Based Threshold Accepting (LBTA) Fu, Eglese, and Li (2005) Tabu Search Heuristic (TS) Pisinger and Ropke (2005) Adaptive Large Neighborhood Search (ALNS)

7 Record-to-Record Travel  A deterministic variant of simulated annealing developed by Dueck (1993)  Framework of RTR (for a minimization problem) Record (R) : Best solution found so far Deviation (D) : Amount of uphill move allowed (D = k% × R) Rule : If Obj (S) < R + D, then solution S is replaced by S

8 Solving the OVRP with RTR Travel  Adapted from RTR for solving large-scale VRPs (Li, Golden, and Wasil 2005) to solve the open vehicle routing problem (ORTR)  Features of ORTR Fixed-length neighbor list with 20 customers (tradeoff between running time and solution quality) Sweep algorithm to generate an initial solution with a minimum number of vehicles Combine two routes (if possible) to reduce the total number of vehicles even if the total distance increases

9 Computational Results  Benchmark data sets 16 test problems C1 to C14 from Christofides et al. (1979) F11, F12 from Fisher (1994) 50 to 199 customers 7 problems have a route-length restriction  ORTR coded in Java Athlon 1 GHz, 256 MB RAM, Linux

10 Computational Results  Illustrative results Problem Kmin Minimize Vehicles with Least Distance C TSF C ALNS 25K, ALNS 50K, ORTR C TSF (6 vehicles) C ALNS 25K, ALNS 50K, ORTR (11 vehicles) F ORTR Problem Kmin Minimize Vehicles with Least Number of Vehicles C TSF C BR, BATA, LBTA (11 vehicles) C TSF (6 vehicles) C TSR (11 vehicles) F ORTR

11 Computational Results  Aggregate results for top four procedures TSRALNS 25KALNS 50KORTR Total Number of Vehicles (sum of Kmin = 147) Total Distance10,12310,19910,19410,191 Time (s)6,34713,35022,2001,756

12 Computational Results  When the number of vehicles is minimized ALNS 50K generated best solution to 9 problems (56%) ALNS 25K 7 problems (44%) ORTR 5 problems (31%)  When the total distance traveled is minimized TSR generated best solution to 5 problems (31%) ORTR 4 problems (25%)

13 Computational Results  ORTR solutions Problem C2, n = 75, solution value =

14 Computational Results  ORTR solutions Problem C14, n = 100, solution value =

15 Computational Results  ORTR solutions Problem F12, n = 134, solution value =

16 Large-Scale Test Problems  New test problems 8 problems LSVRPs from Golden et al. (1998) 200 to 480 customers No route-length restriction Geometric symmetry Customers in concentric circles around the depot Visually estimate solutions

17 Large-Scale Test Problems  Visually estimated solutions Problem O1, n = 200, estimated solution value =

18 Large-Scale Test Problems  Visually estimated solutions Problem O8, n = 480, estimated solution value =

19 Computational Results  Results for ORTR on new test problems % Improvement over ProblemnC Kmin ORTRTime(s)Estimated Solution O O O O O O O O Total Vehicles6868 Average1.57 C is vehicle capacity Estimated solutions used 72

20 Large-Scale Test Problems  ORTR solutions Problem O1, n = 200, estimated solution value =

21 Large-Scale Test Problems  ORTR solutions Problem O1, n = 480, estimated solution value =

22 Conclusions  Increased interest in the OVRP in last five years Contractors used to deliver packages and newspapers Wide variety of new algorithms to solve problems  Three algorithms were accurate Adaptive large neighborhood search (ALNS 25K, ALNS 50K) Tabu search (TSR) Record-to-record travel (ORTR)  Generated eight large-scale test problems ORTR found good solutions in a few minutes